The algebraic K-theory of $k[\operatorname{SL}_2(\mathbb{F}_q)]$

Abstract: We compute via trace methods the higher algebraic $K$-theory of the group ring $k[\operatorname{SL}_2(\mathbb{F}_q)]$, as well as the related groups $\operatorname{PSL}_2(\mathbb{F}_q)$, $\operatorname{PGL}_2(\mathbb{F}_q)$, and $\operatorname{GL}_2(\mathbb{F}_q)$, where $k$ is a perfect field of characteristic $p$ and $q=p<sup>r$. At the core of the computation is the algebraic $K$-theory of the group ring of the Sylow $p$-subgroup, $k[C_p<sup>r]$, which we determine via a theorem of Lück--Reich--Rognes--Varisco on cyclic assembly for topological cyclic homology. In the process, we reprove the cyclic assembly result in the language of Nikolaus--Scholze, analyse assembly for smaller families of subgroups, and develop further tools for computing topological cyclic homology of group rings.

• The paper provides explicit computations of p-torsion in the higher algebraic K-groups of modular group rings for groups including SL₂(Fq) and GL₂(Fq).
• It employs trace methods and topological cyclic homology to reduce complex K-theory calculations to module-theoretic computations via Sylow p-subgroups.
• The work establishes a computational framework using cyclic assembly and induction theory, with significant implications for modular representation and stable homotopy theory.

Algebraic $K$-Theory of Modular Group Rings of $\operatorname{SL}_2(\mathbb{F}_q)$ and Related Groups

Overview

The paper "The algebraic K-theory of $k[\operatorname{SL}_2(\mathbb{F}_q)]$" (2606.21421) delivers explicit computations of the higher algebraic $K$-groups of modular group rings $k[G]$ for $$G \in \{\operatorname{SL}_2(\mathbb{F}_q), \operatorname{PSL}_2(\mathbb{F}_q), \operatorname{PGL}_2(\mathbb{F}_q), \operatorname{GL}_2(\mathbb{F}_q)\}$$ where $k$ is a perfect field of characteristic $p$ with $q= p^r$. The computations isolate the $p$-torsion in the higher $\operatorname{SL}_2(\mathbb{F}_q)$0-theory and establish a direct connection to topological cyclic homology ($\operatorname{SL}_2(\mathbb{F}_q)$1), leveraging trace methods, cyclic and $\operatorname{SL}_2(\mathbb{F}_q)$2-local assembly phenomena, and explicit analyses of Sylow $\operatorname{SL}_2(\mathbb{F}_q)$3-subgroups.

Modular Representation Theory and Higher $\operatorname{SL}_2(\mathbb{F}_q)$4-Theory of Group Rings

The $\operatorname{SL}_2(\mathbb{F}_q)$5-theory of modular group rings is crucial in the intersection of representation theory, homological algebra, and stable homotopy theory. For $\operatorname{SL}_2(\mathbb{F}_q)$6 and $\operatorname{SL}_2(\mathbb{F}_q)$7, classical results of Brauer and others provide concrete computations: $\operatorname{SL}_2(\mathbb{F}_q)$8 has rank equal to the number of $\operatorname{SL}_2(\mathbb{F}_q)$9-conjugacy classes of $k[\operatorname{SL}_2(\mathbb{F}_q)]$0-elements, and $k[\operatorname{SL}_2(\mathbb{F}_q)]$1 is the abelianization of the group of units.

However, there exists a paucity of explicit descriptions of higher $k[\operatorname{SL}_2(\mathbb{F}_q)]$2-groups $k[\operatorname{SL}_2(\mathbb{F}_q)]$3 for $k[\operatorname{SL}_2(\mathbb{F}_q)]$4 in the general modular case. The computational bottleneck involves understanding contributions from non-semisimple and nilpotent phenomena in $k[\operatorname{SL}_2(\mathbb{F}_q)]$5 when $k[\operatorname{SL}_2(\mathbb{F}_q)]$6. The approach here circumvents direct calculation in $k[\operatorname{SL}_2(\mathbb{F}_q)]$7-theory by shifting to homotopical invariants, specifically $k[\operatorname{SL}_2(\mathbb{F}_q)]$8, which retains all nontrivial $k[\operatorname{SL}_2(\mathbb{F}_q)]$9-torsion $K$0-theory information for finite-dimensional algebras over perfect fields.

Main Results: Structure of $K$1

The principal algebraic result gives, for $K$2,

$K$3

where $K$4 denotes the Jacobson radical and $K$5 is the semisimple Artin-Wedderburn quotient. The invariants $K$6 are described as coefficients in explicit Hilbert series

$K$7

with the multiplicity $K$8 depending on the group structure:

• $K$9 for $k[G]$0,
• $k[G]$1 for $k[G]$2,
• $k[G]$3 for $k[G]$4,
• $k[G]$5 for $k[G]$6.

The semi-simple quotient contributes no $k[G]$7-torsion in higher degrees due to unique $k[G]$8-divisibility (\emph{cf.} Hiller's theorem, Quillen's calculation for finite fields).

Strong result: for these families, all $k[G]$9-torsion in higher $$G \in \{\operatorname{SL}_2(\mathbb{F}_q), \operatorname{PSL}_2(\mathbb{F}_q), \operatorname{PGL}_2(\mathbb{F}_q), \operatorname{GL}_2(\mathbb{F}_q)\}$$0-theory is accounted for by trace and assembly computations in $$G \in \{\operatorname{SL}_2(\mathbb{F}_q), \operatorname{PSL}_2(\mathbb{F}_q), \operatorname{PGL}_2(\mathbb{F}_q), \operatorname{GL}_2(\mathbb{F}_q)\}$$1, reducing the problem to explicit module-theoretic calculations determined by the Sylow $$G \in \{\operatorname{SL}_2(\mathbb{F}_q), \operatorname{PSL}_2(\mathbb{F}_q), \operatorname{PGL}_2(\mathbb{F}_q), \operatorname{GL}_2(\mathbb{F}_q)\}$$2-subgroups and their normalizers.

Trace Methods and Topological Cyclic Homology

The heart of the computational strategy is the application of trace invariants—specifically, the comparison map $$G \in \{\operatorname{SL}_2(\mathbb{F}_q), \operatorname{PSL}_2(\mathbb{F}_q), \operatorname{PGL}_2(\mathbb{F}_q), \operatorname{GL}_2(\mathbb{F}_q)\}$$3 for finite-dimensional $$G \in \{\operatorname{SL}_2(\mathbb{F}_q), \operatorname{PSL}_2(\mathbb{F}_q), \operatorname{PGL}_2(\mathbb{F}_q), \operatorname{GL}_2(\mathbb{F}_q)\}$$4 over $$G \in \{\operatorname{SL}_2(\mathbb{F}_q), \operatorname{PSL}_2(\mathbb{F}_q), \operatorname{PGL}_2(\mathbb{F}_q), \operatorname{GL}_2(\mathbb{F}_q)\}$$5 is an isomorphism on $$G \in \{\operatorname{SL}_2(\mathbb{F}_q), \operatorname{PSL}_2(\mathbb{F}_q), \operatorname{PGL}_2(\mathbb{F}_q), \operatorname{GL}_2(\mathbb{F}_q)\}$$6-torsion. $$G \in \{\operatorname{SL}_2(\mathbb{F}_q), \operatorname{PSL}_2(\mathbb{F}_q), \operatorname{PGL}_2(\mathbb{F}_q), \operatorname{GL}_2(\mathbb{F}_q)\}$$7 is far more computable due to the compatibility of its homotopical and cyclical structures, permitting induction along subgroup families and localization at $$G \in \{\operatorname{SL}_2(\mathbb{F}_q), \operatorname{PSL}_2(\mathbb{F}_q), \operatorname{PGL}_2(\mathbb{F}_q), \operatorname{GL}_2(\mathbb{F}_q)\}$$8.

Key in this approach is the use of cyclic assembly, as developed by Lück–Reich–Rognes–Varisco, which realizes $$G \in \{\operatorname{SL}_2(\mathbb{F}_q), \operatorname{PSL}_2(\mathbb{F}_q), \operatorname{PGL}_2(\mathbb{F}_q), \operatorname{GL}_2(\mathbb{F}_q)\}$$9 (for $k$0 connective, $k$1 finite) as a colimit over the family of cyclic subgroups. The author re-proves (and in some sense generalizes) the cyclic assembly result in the language of Nikolaus–Scholze's cyclotomic spectra, extending applicability to possibly infinite groups after $k$2-completion.

Explicit Calculations for Elementary Abelian Sylow $k$3-Subgroups

The most critical computational input is the analysis of $k$4, leveraging isomorphisms

$k$5

reducing the computation to that of truncated polynomial algebras. The explicit computation yields, for $k$6,

$k$7

with the same Hilbert series as above.

The group-theoretic structure of these modules is elucidated by analyzing the action of Singer cycles and more generally of subgroups of $k$8 acting freely on $k$9. Notably, in contexts where such an action is free, the resulting $p$0-module exhibits a free representation structure.

Reduction to Normalizers and Borel Subgroups

For the groups $p$1 above, passage from the $p$2-Sylow subgroup to the entire group is executed via explicit analysis of the normalizer (the Borel subgroup). In many cases, particularly for $p$3 and $p$4, the normalizer is a Frobenius group, and the author establishes a pushout formula for $p$5 in the presence of Frobenius structures, analogous to classical statements in modular representation theory.

All calculations for the Borel subgroups are explicitly reduced to earlier Hilbert series computations, with group ring decompositions facilitating passage to block summands in the Artin–Wedderburn context.

$p$6-Locality and Induction Theory

Beyond the calculations for the specific groups in question, the paper demonstrates, both via stable module categories and via equivariant homotopy (Appendix A), that for a wide class of group rings over perfect fields of characteristic $p$7, higher algebraic $p$8-theory is $p$9-local: it is determined by values on $q= p^r$0-local subgroups. This is formalized using spectral Mackey functors and Brown's complex of nontrivial $q= p^r$1-subgroups, yielding explicit split exact sequences relating the $q= p^r$2-groups, and extending induction-theoretic ideas familiar from group cohomology to the stable and categorical context.

Implications and Future Directions

These results resolve the explicit form of higher $q= p^r$3-theory for the modular group rings of the most fundamental finite groups of Lie type $q= p^r$4 and establish a computational paradigm for a much broader class via assembly and trace methods. The explicit reduction to $q= p^r$5-local subgroups and the translation to $q= p^r$6 foster the prospect of future extensions to more complex reductive groups and systematic calculations in equivariant algebraic $q= p^r$7-theory, particularly in modular representation theory and its interface with stable homotopy theory.

The computational infrastructure—particularly the assembly framework, explicit handling of Frobenius normalizers, and induction theory—can feed into practical algorithms for higher $q= p^r$8-theory and $q= p^r$9, impacting questions in automorphic forms, the study of buildings and their homology, and advances in trace invariants beyond the group ring context. Furthermore, the categorical perspective on such computations supplies a blueprint for understanding the defect bases and systematic projectivity in generalized cohomology theories for discrete groups.

Conclusion

This work clarifies the $p$0-torsion profile of higher algebraic $p$1-theory for modular group rings of classical groups $p$2, $p$3, and relatives, by exploiting an overview of trace invariants, cyclic and $p$4-local assembly, and detailed algebraic and homotopical analyses of Sylow subgroups and their normalizers. The technical framework here not only answers explicit computation questions but also sketches a road-map for future advances in the algebraic $p$5-theory of finite group algebras, the structure of assembly and descent, and the role of $p$6-local subgroups in categorical and spectral algebra.